On Generalized Fibonacci Numbers

TL;DR

This paper derives a formula for the nth term of k-generalized Fibonacci-like sequences, analyzes their ratio limits, and extends results to related sequences, revealing convergence to roots of specific equations.

Contribution

It introduces a new formula for k-generalized Fibonacci-like numbers and extends the analysis to Horadam and Horadam-like sequences, including their ratio limits.

Findings

Ratio of successive terms tends to a root of x + x^{-k} = 2.

Derived explicit formula for the nth term of k-generalized Fibonacci-like sequences.

Extended results to k-generalized Horadam and Horadam-like sequences.

Abstract

We provide a formula for the $n^{th}$ term of the $k$-generalized Fibonacci-like number sequence using the $k$-generalized Fibonacci number or $k$-nacci number, and by utilizing the newly derived formula, we show that the limit of the ratio of successive terms of the sequence tends to a root of the equation $x + x^{-k} = 2$. We then extend our results to $k$-generalized Horadam ($k$GH) and $k$-generalized Horadam-like ($k$GHL) numbers. In dealing with the limit of the ratio of successive terms of $k$GH and $k$GHL, a lemma due to Z. Wu and H. Zhang [8] shall be employed. Finally, we remark that an analogue result for $k$-periodic $k$-nary Fibonacci sequence can also be derived.